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Write the equation in vertex form for the parabola with vertex (0,−8)(0,-8) and focus (0,−7)(0,-7).\newlineSimplify any fractions.\newline______

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Q. Write the equation in vertex form for the parabola with vertex (0,−8)(0,-8) and focus (0,−7)(0,-7).\newlineSimplify any fractions.\newline______
  1. Identify Vertex Form: Identify the vertex form of a parabola.\newlineVertex form: y=a(x−h)2+ky = a(x-h)^2 + k
  2. Given Vertex and Focus: Given vertex (h,k)=(0,−8)(h,k) = (0,-8) and focus (0,−7)(0,-7).\newlineSince the focus is above the vertex, the parabola opens upwards.
  3. Calculate Distance for 'a': Calculate the distance between the vertex and focus to find the value of 'a'.\newlineDistance = ∣k−focus y-coordinate∣=∣−8−(−7)∣=1|k - \text{focus y-coordinate}| = |-8 - (-7)| = 1
  4. Find 'a' Value: Use the distance to find 'a'.\newlineThe distance is equal to 14a\frac{1}{4a}, so 1=14a1 = \frac{1}{4a}.\newlineSolve for 'a': 4a=14a = 1, a=14a = \frac{1}{4}.
  5. Substitute Values and Simplify: Substitute aa, hh, and kk into the vertex form equation.y=(14)(x−0)2+(−8)y = \left(\frac{1}{4}\right)(x-0)^2 + (-8)Simplify the equation.y=(14)x2−8y = \left(\frac{1}{4}\right)x^2 - 8

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